In the S-category $P$ (with compact-open strong shape mappings, cf. §1, instead of continuous mappings, and arbitrary finite-dimensional separable metrizable spaces instead of finite polyhedra) there exists according to [1], [2] an S-duality. The S-dual $DX,X=(X,n)\in P$, turns out to be of the same weak homotopy type as an appropriately defined functional dual $\overline{{\left(S^0\right)}^X}$ (Corollary 4.9). Sometimes the functional object $\overline{X^Y}$ is of the same weak homotopy type as the “real” function space $X^Y$ (§5).

If $X$ is a $n$-dim polyhedran, then using geometric techniques, we construct groups $H_p(X{)}_{\Delta }$ and $H^p(X{)}_{\Delta }$ such that there are natural isomorphisms $H^p(X{)}_{\Delta }\simeq H_{n-p}\left(X\right)$ and $H_p(X{)}_{\Delta }\simeq H^{n-p}\left(X\right)$ which induce an intersection pairing. These groups give a geometric interpretation of two spectral sequences studied by Zeeman and allow us to prove a conjecture of Zeeman about them.